R E S E A R C H
Open Access
On parametric implicit vector variational
inequality problems
A Farajzadeh
1and S Plubteing
2**Correspondence:
2Department of Mathematics,
Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand
Full list of author information is available at the end of the article
Abstract
Recently Huanget al.(Math. Comput. Model. 43:1267-1274, 2006) introduced a class of parametric implicit vector equilibrium problems (for short PIVEP) and they presented some existence results for a solution of PIVEP. Also, they provided two theorems about upper and lower semi-continuity of the solution set of PIVEP in a locally convex Hausdorff topological vector space. The paper extends the
corresponding results obtained in the setting of topological vector spaces with mild assumptions and removing the notion of locally non-positiveness at a point and lower semi-continuity of the parametric mapping.
Keywords: topological vector space; equilibrium problem; KKM mapping
1 Introduction and preliminaries
Equilibrium problems have been extensively studied in recent years, the origin of which can be traced back to Takahashi [, Lemma ], Blum and Oettli [], and Noor and Oettli []. It is well known that vector equilibrium problems provide a unified model for several classes of problems, for example, vector variational inequality problems, vector comple-mentarity problems, vector optimization problems, and vector saddle point problems; see [–] and the references therein. In , Huanget al.[] considered the implicit vector equilibrium problem (for short IVEP) which consists of findingx∈Esuch that
fg(x),y∈/–intC(x), ∀y∈E,
wheref :E×E→Yandg:E→E, are mappings,XandYare two Hausdorff topological vector spaces,Eis a nonempty closed convex subset ofXandC:E→Y be a set-valued
mapping such that for anyx∈E,C(x) is a closed and convex cone withC(x)∩–C(x) ={}, that is pointed, with nonempty interior. They continued their research and introduced the parametric implicit vector equilibrium problem, which consists of findingx∗∈K(λ), for each given (λ,)∈×such that
f,gx∗,y∈/–intCx∗, ∀y∈K(λ),
wherei (i= , ) are Hausdorff topological vector spaces (the parametric spaces),K:
→Xa set-valued mapping such that for anyλ∈,K(λ) is a nonempty, closed and
convex subset ofXwith K() =
λ∈K(λ)⊆E andf :××E→Y. They
ob-tained some existence results for a solution of PIVEP and further they studied upper and
lower semi-continuity of the solution of PIVEP in locally convex Hausdorff topological vector spaces. This paper is motivated and inspired by the recent paper [] and its aim is to extend the results to the setting of Hausdorff topological vector spaces with mild as-sumptions and removing the condition of being locally non-positive at a point has been applied in Proposition . of [] and lower semi-continuity of the parametric mapping used in Theorem . of []. More precisely, we first establish an existence result for a so-lution of IVEP and then by using it we will deal with the behavior of the soso-lution set of PIVEP when the parameters (λ,) start to change. In fact we will show that the solution set as a mappingS:×→Xis upper semi-continuous and lower semi-continuous
under special conditions. In the rest of this section we recall some definitions and results that we need in the next section.
A subsetPofYis called a pointed and convex cone if and only ifP+P⊆P,tP⊆P, for allt≥, andP∩–P={}. The domain of a set-valued mappingW:X→Y is defined as
D(W) ={x∈X:W(x)=∅}and its graph is defined as
Graph(W) =(x,z)∈X×Y:z∈W(x).
AlsoWis said to be closed if its graph, that is,Graph(W), is a closed subset ofX×Y. A set-valued mappingT:X→Y is called upper semi-continuous (u.s.c.) atx∈Xif for every
open setV containingT(x) there exists an open setUcontainingxsuch thatT(u)⊆V, for allu∈U. The mappingTis said to be lower semi-continuous (l.s.c.) if for every open setVwithT(x)∩V =∅there exists an open setUcontainingxsuch thatT(u)∩V =∅.
The mappingT is continuous atxif it is both u.s.c. and l.s.c. atx. Moreover,T is u.s.c. (l.s.c.) onXifT is u.s.c. (l.s.c.) at each point ofX.
We need the following lemma in the sequel.
Lemma .([]) Let X and Y be topological spaces and T:X→Y be a mapping.The
following statements are true:
(i) If for anyx∈X,T(x)is compact,thenTis u.s.c.atx∈Xif and only if for any net {xi} ⊆Xsuch thatxi→xand for everyyi∈T(xi),there existy∈T(x)and a subnet
{yj}of{yi}such thatyj→y.
(ii) T is l.s.c.atx∈Xif and only if for any net{xi} ⊆Xwithxi→xand for anyy∈T(x),
there exists a net{yi}such thatyi∈T(xi)andyi→y.
Definition .([, ]) LetXbe a topological vector space. A mappingF:K⊆X→Xis
said to be a KKM mapping, if, for any finite setA⊆K,
coA⊆F(A) =
x∈A
F(x),
wherecoAdenotes the convex hull ofA.
The following lemma plays a crucial rule in this paper.
Lemma .([]) Let K be a nonempty subset of a topological vector space X and F:K→
Xbe a KKM mapping with closed values in K.Assume that there exists a nonempty
2 Main results
The next result provides an existence result for a solution of IVEP.
Theorem . Let K be closed convex subset of a t.v.s.X and f:K×K→Y and g:K→K be two mappings.If the following assumptions are satisfied:
(a) f(g(x),x) /∈–intC(x),∀x∈K,
(b) the mappingx→f(g(x),y)is continuous,for ally∈K, (c) for eachx∈K,the set{y∈K:f(g(x),y)∈–intC(x)}is convex, (d) the mappingW:K→Y defined byW(x) =Y\(–intC(x))is closed,
(e) there exist subsetsMandNofK,compact convex and compact,respectively,such that for allx∈K\Nthere isy∈Msuch thatf(g(x),y)∈–intC(x),
then the solution set of IVEP is nonempty and compact.
Proof Define the set-valued mappingF:K→Kby
F(y) =x∈K:fg(x),y∈/–intC(x).
We show that F satisfies all the assumptions of Lemma .. By (b) and (d), F(y) is a closed subset ofK for ally∈K. It follows from (c) and (a) thatF is a KKM mapping. Indeed, on the contrary of the assertion if there existy,y, . . . ,yninKandz= ni=λiyi∈
co{y,y, . . . ,yn}\
n
i=F(yi), thenf(g(z),yi)∈–intC(z) and so by (c) we deduce that
fg(z),z∈–intC(z),
which is a contradiction (by (a)). ThenFis a KKM mapping. Also, it is obvious from (e) thaty∈MF(y)⊆Nand soy∈MF(y) is compact (note thatF(y) is closed for eachy∈Y
andMis compact). Hence by Lemma . there existsx∈Ksuch that
x∈
x∈K
F(x)
and it is easy to see that the solution set of IVEP is equal to the setx∈KF(x) and hence ¯
xis a solution of IVEP and further it is compact (notex∈KF(x)⊆x∈MF(x)⊆N) and
hence the proof is complete.
We note that ifg is continuous andf is continuous with respect to the first variable then the mappingx→f(g(x),y) is continuous and so condition (b) holds while the simple exampleg(x) = ifxis rational, andg(x) = ifxis irrational, andf(x,y) = , forxrational, andf(x,y) = ifxis irrational, shows that it is easy to check that the mappingx→f(g(x),y) is continuous; nevertheless, neithergnorf is continuous, which shows that the converse does not hold in general. Moreover, in the example, if we takeK = [, ] thenf andg
satisfy all the assumptions of Theorem . and so the solution set of IVEP is nonempty and compact but the example cannot fulfill all the conditions of Proposition . in []. Hence Theorem . extends Proposition . in []. Also one can easily see theC-convexity off
at the second variable, that is, for eachx∈K,
which implies condition (c) of Theorem ., while if we takeX= and letK be any nonempty convex and compact subset ofXand definef(x,y) = –y, for allx,y∈K and
we letgbe an arbitrary mapping, we take the examplef(x,y) = –y, forx,y∈Kandgan
arbitrary mapping, then this example fulfills condition (c) (note that it satisfies all the as-sumptions of Theorem .) butfis not convex at the second variable and hence condition (c) improves condition () in Proposition . of [].
Definition .([]) A mappingf :E×E→Yis said to be locally non-positive atx∈E
with respect to a mappingg:E→Eif there exist a neighborhoodV(x) ofxand a point
z∈E∩intV(x) such that
fg(x),z∈–C(x), ∀x∈E∩∂V(x),
where ∂V(x) is the boundary ofV(x). In the case thatg is the identity mapping, the
mappingf is called locally non-positive atx∈E.
The following corollary is an extension of Proposition . in [] for topological vector spaces. Furthermore, the condition thatf is locally non-positive atx∈Khas been omit-ted.
Corollary . Let K be a nonempty closed and convex subset of a Hausdorff topological vector space X and let f :K×K→Y and C:K→Y be two mappings such that:
(a) f(g(x),x) = ,∀x∈K,
(b) the mappingx→f(g(x),y)is continuous,for ally∈K, (c) for eachx∈Kthe mappingy→f(g(x),y)isC(x)-convex,
(d) the mappingW:K→Y defined byW(x) =Y\(–intC(x))is closed,
(e) there exist a nonempty compact and convex subsetDofK∩V(x)andy∈Dsuch that for allx∈(K∩V(x))\D
fg(x),y∈–intC(x).
Then IVEP has a solution in the neighborhood V(x)of x,that is,there exists x∗∈(K∩ V(x))such that
fgx∗,y∈/intCx∗, ∀y∈K.
Moreover,the solution set is a compact subset of K∩V(x).
Proof There is neighborhood U ofxsuch that coU ⊆V(x) (see, for example, []). Hence by Theorem ., IVEP has a solution onB=K∩co(coU∪ {y}). Then there ex-istsx∗∈Bsuch that
fgx∗,y∈–/ intCx∗, ∀y∈B.
We claim that
Indeed, if the sentence is not true then there isy∈Kso that
fgx∗,y∈–intCx∗.
Putyt=x∗+t(y–x∗), fort> . It is clear thatyt∈B, fortthat is small enough. Then by
condition (c) we have
fgx∗,yt
∈( –t)fgx∗,y+tfgx∗,x∗∈–intCx∗+ =–intCx∗,
which is a contradiction. Hencex∗ is a solution of IVEP. The second part follows from
condition (e). This completes the proof.
The next theorem is an extension of Theorems ., . and Corollary . in [] with mild assumptions for mappings which do not need to satisfy the locally non-positive condition. In fact this condition has been removed.
Theorem . Let F:×E×E→Y and g:E→E be two mappings.If the following assumptions hold:
(i) K:→Eis a continuous mapping with nonempty convex compact values;
(ii) (ε,x,y)→F(ε,g(x),y)is continuous;
(iii) the set{y:F(ε,g(x),y)∈–intC(x)}is convex andF(ε,g(x),x) = ,for each (,x)∈×E;
(iv) the mappingW:E→Ydefined byW(x) =Y\–intC(x)is closed;
then
(i) for each(λ,ε)∈×,the solution set
S(λ,ε) =x∈K(λ) :Fε,g(x),y∈/–intC(x),∀y∈K(λ),
is nonempty and compact,
(ii) the solution set mappingS:×→Xdefined by
(λ,ε)→S(λ,ε)
is continuous.
Proof The first part, that is, (i), follows from Theorem . by taking, for each (λ,ε)∈× ,M=N=K(λ) and definingf(x,y) =F(ε,x,y), for all (x,y)∈K(λ)×K(λ). To prove (ii),
let{(λi,i)}i∈I⊆×be a net with (λi,i)→(λ,) andzi∈S(λi,i)⊂K(λi). SinceKis
u.s.c.,λi→λandzi∈K(λi), using Lemma .(i), there existz∈K(λ) and a subnet{zij}of
{zi}which converges toz. So
Fij,g(zij),y
∈Y\–intC(zij)
=W(zij), ∀y∈K(λij). ()
We claim thatz∈S(λ,) (note that if we show the claim then according to Lemma . the mappingSwill be an u.s.c.). If the claim is not true then there isy∈K(λ) such that
So, sinceKis l.s.c., there exists netwj∈K(λj) such thatwj→y. Then it follows from ()
that
Fij,g(zij),wj
∈W(zij),
and so by (ii) and (iv) we get
F,g(z),y∈Y\–intC(z) =W(z),
which is contradicted by () and so z ∈ S(λ,). It follows from xij ∈ S(λij,ij) that f(ij,g(xij),ij) /∈–intC(xij) and sof(ij,g(xij),yij)∈W(xij) and by the closedness ofW we
getf(,g(x),y)∈W(x), which is a contradiction, and then the solution set mappingSis u.s.c. Now we show thatSis l.s.c. Let{(λi,i)}i∈I⊆×be a net with (λi,i)→(λ,)
andzan arbitrary element ofS(λ,)⊆K(λ). Put={V:Vis a neighborhood ofz}(note by the relationVW if and only ifV⊇W, the setis a directed set). Then for each (V,i)∈×I, there is a closed and convex neighborhoodHV,iofzsuch thatHV,i⊂V(see
[]) and so it follows from Theorem . that there iszi∈HV,i∩K(λi) such that
Fi,g(zi),y
/
∈–intC(zi), ∀y∈HV,i∩K(λi). ()
Now if there existsy∈K(λi) such that
Fi,g(zi),y
∈–intC(zi),
it follows fromF(i,g(zi),zi) = (see condition (i)) and (iii) that
Fi,g(zi),zi+t(y–zi)
∈–intC(zi), ∀t∈[, ],
which is a contradiction, fort∈[, ] small enough, by () (noteHV,iis an open set and
zi∈HV,i). So
Fi,g(zi),y
/
∈–intC(zi), ∀y∈K(λi),
and hencezi∈S(i,λi). Consequently, for each point (V,i)∈×Ithere iszi∈S(i,λi), and
sozi→z. Hence it follows from Lemma .(ii) thatSis l.s.c. and the proof is completed.
Inspired by the proof of the second part of the previous theorem we can deduce the lower semi-continuity of the solution set mapping. Indeed the next theorem is an improvement of Theorem . in [] without using the lower semi-continuity of the mappingK:→
E; its proof is similar to the proof presented for the second part of Theorem . and so
we omit the proof.
Theorem . Let F:×E×E→Y and g :E→E be two mappings. For a given
(i) K:U(λ)→Xis a mapping with nonempty convex and compact values;
(ii) mapping(ε,x,y)→F(ε,g(x),y)is continuous,for each(ε,x,y)∈M()×E×E;
(iii) for eachx∈Ethe mappingy→F(ε,g(x),y)isC(x)-convex andF(ε,g(x),x) = ,for each(,x)∈M()×E;
(iv) the mappingW:E→Ydefined byW(x) =Y\–intC(x)is closed.
Then
(i) S(λ,ε),for each(λ,ε)∈U(λ)×M(ε)is nonempty and compact; (ii) the mappingSis l.s.c.at(λ,ε).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
This work presented have was carried out in collaboration between all authors. The main idea of this paper was proposed by AF and SP prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Razi University, Kermanshah, 67149, Iran.2Department of Mathematics, Faculty of Science,
Naresuan University, Phitsanulok, 65000, Thailand.
Received: 13 August 2013 Accepted: 28 March 2014 Published:02 May 2014
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10.1186/1029-242X-2014-151
